
# Sparkling wine sales


import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns


df_1=pd.read_csv(r'C:\Users\Sandeep Immadi\Downloads\Sparkling.csv')
df_1.head()


df_2 = pd.read_csv(r'C:\Users\Sandeep Immadi\Downloads\Sparkling.csv',parse_dates=True,squeeze=True,index_col=0)


print(df_2.head())
print('\n')
print(df_2.tail())

YearMonth
1980-01-01    1686
1980-02-01    1591
1980-03-01    2304
1980-04-01    1712
1980-05-01    1471
Name: Sparkling, dtype: int64


YearMonth
1995-03-01    1897
1995-04-01    1862
1995-05-01    1670
1995-06-01    1688
1995-07-01    2031
Name: Sparkling, dtype: int64


date=pd.date_range(start='1/1/1980',end='8/1/1995',freq='M')
date

DatetimeIndex(['1980-01-31', '1980-02-29', '1980-03-31', '1980-04-30',
               '1980-05-31', '1980-06-30', '1980-07-31', '1980-08-31',
               '1980-09-30', '1980-10-31',
               ...
               '1994-10-31', '1994-11-30', '1994-12-31', '1995-01-31',
               '1995-02-28', '1995-03-31', '1995-04-30', '1995-05-31',
               '1995-06-30', '1995-07-31'],
              dtype='datetime64[ns]', length=187, freq='M')


df_1['Time_Stamp'] = pd.DataFrame(date,columns=['Month'])
df_1.head()


df = df_1.set_index('Time_Stamp',)
df.drop('YearMonth',axis=1,inplace=True)
df.head()


# Plotting the time series for 'Sparkling' wine sales


df.plot(figsize=(20,8))
plt.grid()
plt.title("Sparkling Sales")
plt.show()


# Checking null values for 'Sparkling' wine sales
df.isnull().sum()

Sparkling    0
dtype: int64


# Descriptive Analysis of 'Sparkling' Wine Sales
df.describe()


# Plotting Boxplot for year wise
from pylab import rcParams
rcParams['figure.figsize'] = 20,8


sns.boxplot(x=df.index.year,y=df['Sparkling'])
plt.grid();


# Plotting Boxplot for Month wise

sns.boxplot(x=df.index.month_name(),y=df['Sparkling'])
plt.grid();


# Plotting the Sparkling wine year month wise sales - Line plot


monthly_sales_across_years=pd.pivot_table(df,values='Sparkling',columns=df.index.month_name(),index=df.index.year)
monthly_sales_across_years


monthly_sales_across_years.plot()
plt.grid()
plt.legend(loc='best');


# Plotting a month plot to check the sales in different years and withing different month across


## importing the month plot from statsmodels
from statsmodels.graphics.tsaplots import month_plot


month_plot(df['Sparkling'],ylabel='Sparkling Sales')
plt.grid();


# Plot the Empirical Cumulative Distribution


# statistics
from statsmodels.distributions.empirical_distribution import ECDF
plt.figure(figsize = (18, 8))
cdf = ECDF(df['Sparkling'])
plt.plot(cdf.x, cdf.y, label = "statmodels");
plt.grid()
plt.xlabel('Sales');


# Plot the average Sparkling Wine Sales per month and the month on month percentage change of Sparkling Wine Sales


# group by date and get average Sparkling wine Sales, and precent change
average = df.groupby(df.index)["Sparkling"].mean()
pct_change = df.groupby(df.index)["Sparkling"].sum().pct_change()
fig, (axis1,axis2) = plt.subplots(2,1,sharex=True,figsize=(15,8))
# plot average Sparkling wine Sales over time(year-month)
ax1 = average.plot(legend=True,ax=axis1,marker='o',title="Average Sparkling Wine Sales",grid=True)
ax1.set_xticks(range(len(average)))
ax1.set_xticklabels(average.index.tolist())
# plot precent change for Rpse Wine Sales over time(year-month)
ax2 = pct_change.plot(legend=True,ax=axis2,marker='o',colormap="summer",title="Sparkling wine Sales Percent Change",grid=True)


# Decomposing the time series into Additive decomposition and plot


from statsmodels.tsa.seasonal import seasonal_decompose
decomposition = seasonal_decompose(df['Sparkling'],model='Additive')
decomposition.plot();


trend= decomposition.trend
seasonality=decomposition.seasonal
residual=decomposition.resid
print('Trend','\n',trend.head(12),'\n')
print('Seasonality','\n',seasonality.head(12),'\n')
print('Residual','\n',residual.head(12),'\n')

Trend 
 Time_Stamp
1980-01-31            NaN
1980-02-29            NaN
1980-03-31            NaN
1980-04-30            NaN
1980-05-31            NaN
1980-06-30            NaN
1980-07-31    2360.666667
1980-08-31    2351.333333
1980-09-30    2320.541667
1980-10-31    2303.583333
1980-11-30    2302.041667
1980-12-31    2293.791667
Name: trend, dtype: float64 

Seasonality 
 Time_Stamp
1980-01-31    -854.260599
1980-02-29    -830.350678
1980-03-31    -592.356630
1980-04-30    -658.490559
1980-05-31    -824.416154
1980-06-30    -967.434011
1980-07-31    -465.502265
1980-08-31    -214.332821
1980-09-30    -254.677265
1980-10-31     599.769957
1980-11-30    1675.067179
1980-12-31    3386.983846
Name: seasonal, dtype: float64 

Residual 
 Time_Stamp
1980-01-31           NaN
1980-02-29           NaN
1980-03-31           NaN
1980-04-30           NaN
1980-05-31           NaN
1980-06-30           NaN
1980-07-31     70.835599
1980-08-31    315.999487
1980-09-30    -81.864401
1980-10-31   -307.353290
1980-11-30    109.891154
1980-12-31   -501.775513
Name: resid, dtype: float64


#Decomposing the time series into Additive decomosition and plot


decomposition = seasonal_decompose(df['Sparkling'],model='multiplicative')
decomposition.plot();


trend= decomposition.trend
seasonality=decomposition.seasonal
residual=decomposition.resid
print('Trend','\n',trend.head(12),'\n')
print('Seasonality','\n',seasonality.head(12),'\n')
print('Residual','\n',residual.head(12),'\n')

Trend 
 Time_Stamp
1980-01-31            NaN
1980-02-29            NaN
1980-03-31            NaN
1980-04-30            NaN
1980-05-31            NaN
1980-06-30            NaN
1980-07-31    2360.666667
1980-08-31    2351.333333
1980-09-30    2320.541667
1980-10-31    2303.583333
1980-11-30    2302.041667
1980-12-31    2293.791667
Name: trend, dtype: float64 

Seasonality 
 Time_Stamp
1980-01-31    0.649843
1980-02-29    0.659214
1980-03-31    0.757440
1980-04-30    0.730351
1980-05-31    0.660609
1980-06-30    0.603468
1980-07-31    0.809164
1980-08-31    0.918822
1980-09-30    0.894367
1980-10-31    1.241789
1980-11-30    1.690158
1980-12-31    2.384776
Name: seasonal, dtype: float64 

Residual 
 Time_Stamp
1980-01-31         NaN
1980-02-29         NaN
1980-03-31         NaN
1980-04-30         NaN
1980-05-31         NaN
1980-06-30         NaN
1980-07-31    1.029230
1980-08-31    1.135407
1980-09-30    0.955954
1980-10-31    0.907513
1980-11-30    1.050423
1980-12-31    0.946770
Name: resid, dtype: float64


# Splitting the data into train and test data


train=df[df.index.year < 1991]
test=df[df.index.year >= 1991]


## This is to display multiple data frames from one cell
from IPython.display import display


print('First few rows of Training Data')
display(train.head())
print('Last few rows of Training Data')
display(train.tail())
print('First few rows of Test Data')
display(test.head())
print('Last few rows of Test Data')
display(test.tail())

First few rows of Training Data

Last few rows of Training Data

First few rows of Test Data

Last few rows of Test Data


# Checking the shape of the train and test data
print(train.shape)
print(test.shape)

(132, 1)
(55, 1)


# plotting the graph for train and test set

train['Sparkling'].plot(fontsize=14)
test['Sparkling'].plot(fontsize=14)
plt.grid()
plt.legend(['Training data','Test data'])
plt.show()


train_time=[i+1 for i in range(len(train))]
test_time=[i+133 for i in range(len(test))]
print('Training Time instance','\n',train_time)
print('\n')
print('Test Time instance','\n',test_time)

Training Time instance 
 [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132]


Test Time instance 
 [133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187]


LinearRegression_train = train.copy()
LinearRegression_test = test.copy()


LinearRegression_train['time'] = train_time
LinearRegression_test['time'] = test_time
print('First few rows of Training Data','\n',LinearRegression_train.head(),'\n')
print('Last few rows of Training Data','\n',LinearRegression_train.tail(),'\n')
print('First few rows of Test Data','\n',LinearRegression_test.head(),'\n')
print('Last few rows of Test Data','\n',LinearRegression_test.tail(),'\n')

First few rows of Training Data 
             Sparkling  time
Time_Stamp                 
1980-01-31       1686     1
1980-02-29       1591     2
1980-03-31       2304     3
1980-04-30       1712     4
1980-05-31       1471     5 

Last few rows of Training Data 
             Sparkling  time
Time_Stamp                 
1990-08-31       1605   128
1990-09-30       2424   129
1990-10-31       3116   130
1990-11-30       4286   131
1990-12-31       6047   132 

First few rows of Test Data 
             Sparkling  time
Time_Stamp                 
1991-01-31       1902   133
1991-02-28       2049   134
1991-03-31       1874   135
1991-04-30       1279   136
1991-05-31       1432   137 

Last few rows of Test Data 
             Sparkling  time
Time_Stamp                 
1995-03-31       1897   183
1995-04-30       1862   184
1995-05-31       1670   185
1995-06-30       1688   186
1995-07-31       2031   187


from sklearn.linear_model import LinearRegression


lr = LinearRegression()


lr.fit(LinearRegression_train[['time']],LinearRegression_train['Sparkling'].values)

LinearRegression()


# Plotting Linear Regression Forecast
train_predictions_model1= lr.predict(LinearRegression_train[['time']])
LinearRegression_train['RegOnTime'] = train_predictions_model1
test_predictions_model1= lr.predict(LinearRegression_test[['time']])
LinearRegression_test['RegOnTime']=test_predictions_model1
plt.plot( train['Sparkling'], label='Train')
plt.plot(test['Sparkling'], label='Test')
plt.plot(LinearRegression_train['RegOnTime'],label='Regression On Time_Train_Data')
plt.plot(LinearRegression_test['RegOnTime'], label='Regression On Time_Test Data')
plt.legend(loc='best')
plt.grid();


# Model 1 Evaluation:


### Defining Accuacy Matrix]
from sklearn import metrics
from math import sqrt


# Train - RMSE score


rmse_model1_train = sqrt(metrics.mean_squared_error(train['Sparkling'],train_predictions_model1))
print("For RegressionOnTime forecast on the Train Data, RMSE is %3.3f" %(rmse_model1_train))

For RegressionOnTime forecast on the Train Data, RMSE is 1279.322


# Test - RMSE score


rmse_model1_test = sqrt(metrics.mean_squared_error(test['Sparkling'],test_predictions_model1))
print("For RegressionOnTime forecast on the Test Data, RMSE is %3.3f" %(rmse_model1_test))

For RegressionOnTime forecast on the Test Data, RMSE is 1389.135


# Creating Dataframe


result_DF=pd.DataFrame({'Train_RMSE':[rmse_model1_train],'Test_RMSE':[rmse_model1_test]},index=['RegressionOnTime'])
result_DF


NaiveModel_train = train.copy()
NaiveModel_test = test.copy()


NaiveModel_train['naive'] = np.asarray(train['Sparkling'])[len(np.asarray(train['Sparkling']))-1]
NaiveModel_train['naive'].head()

Time_Stamp
1980-01-31    6047
1980-02-29    6047
1980-03-31    6047
1980-04-30    6047
1980-05-31    6047
Name: naive, dtype: int64


NaiveModel_test['naive'] = np.asarray(train['Sparkling'])[len(np.asarray(train['Sparkling']))-1]
NaiveModel_test['naive'].head()

Time_Stamp
1991-01-31    6047
1991-02-28    6047
1991-03-31    6047
1991-04-30    6047
1991-05-31    6047
Name: naive, dtype: int64


# Plotting Naive Forecast


plt.plot( train['Sparkling'], label='Train')
plt.plot(test['Sparkling'], label='Test')
plt.plot(NaiveModel_train['naive'],label='Naive Forecast On Time_Train_Data')
plt.plot(NaiveModel_test['naive'], label='Naive forecast On Time_Test Data')
plt.legend(loc='best')
plt.title('Naive Forecast')
plt.grid();


# Model 2 Evaluation


# Train - RMSE score
rmse_model2_train = sqrt(metrics.mean_squared_error(train['Sparkling'],NaiveModel_train['naive']))
print("For Naive Model forecast on the Train Data, RMSE is %3.3f" %(rmse_model2_train))

For Naive Model forecast on the Train Data, RMSE is 3867.701


# Test-RMSE score
rmse_model2_test = sqrt(metrics.mean_squared_error(test['Sparkling'],NaiveModel_test['naive']))
print("For Naive Model forecast on the Test Data, RMSE is %3.3f" %(rmse_model2_test))

For Naive Model forecast on the Test Data, RMSE is 3864.279


# Creating Dataframe


result_DF_2=pd.DataFrame({'Train_RMSE':[rmse_model2_train],'Test_RMSE':[rmse_model2_test]},index=['Naive Model'])
result_DF=pd.concat([result_DF,result_DF_2])
result_DF


SimpleAverage_Train=train.copy()
SimpleAverage_Test=test.copy()


SimpleAverage_Train['mean_forecast']= train['Sparkling'].mean()
SimpleAverage_Train.head()


SimpleAverage_Test['mean_forecast']= train['Sparkling'].mean()
SimpleAverage_Test.head()


# Plotting Simple Average Forecast


plt.plot(SimpleAverage_Train['Sparkling'], label='Train')
plt.plot(SimpleAverage_Test['Sparkling'], label='Test')
plt.plot(SimpleAverage_Train['mean_forecast'],label='Simple Average on Training Data')
plt.plot(SimpleAverage_Test['mean_forecast'], label='Simple Average on Test Data')
plt.legend(loc='best')
plt.title("Simple Average Forecast")
plt.grid();


# Model 3 Evaluation


# Train - RMSE score
rmse_model3_train = sqrt(metrics.mean_squared_error(train['Sparkling'],SimpleAverage_Train['mean_forecast']))
print("For Simple Average forecast on the Train Data, RMSE is %3.3f" %(rmse_model3_train))

For Simple Average forecast on the Train Data, RMSE is 1298.484


# Test - RMSE score
rmse_model3_test = sqrt(metrics.mean_squared_error(test['Sparkling'],SimpleAverage_Test['mean_forecast']))
print("For Simple Average forecast on the Test Data, RMSE is %3.3f" %(rmse_model3_test))

For Simple Average forecast on the Test Data, RMSE is 1275.082


# Creating DataFrame


result_DF_3=pd.DataFrame({'Train_RMSE':[rmse_model3_train],'Test_RMSE':[rmse_model3_test]},index=['Simple Average'])
result_DF=pd.concat([result_DF,result_DF_3])
result_DF


MovingAverage = df.copy()
MovingAverage.head()


# Trailing Moving Averages
MovingAverage['Trailing_2'] = MovingAverage['Sparkling'].rolling(2).mean()
MovingAverage['Trailing_4'] = MovingAverage['Sparkling'].rolling(4).mean()
MovingAverage['Trailing_6'] = MovingAverage['Sparkling'].rolling(6).mean()
MovingAverage['Trailing_9'] = MovingAverage['Sparkling'].rolling(9).mean()
MovingAverage.head()


# Plotting Moving Average Forecast


plt.plot(MovingAverage['Sparkling'], label='Train')
plt.plot(MovingAverage['Trailing_2'], label='2 Point Moving Average')
plt.plot(MovingAverage['Trailing_4'], label='4 Point Moving Average')
plt.plot(MovingAverage['Trailing_6'],label = '6 Point Moving Average')
plt.plot(MovingAverage['Trailing_9'],label = '9 Point Moving Average')
plt.legend(loc = 'best')
plt.grid();


# Creating Train and Test set

trailing_MovingAverage_train=MovingAverage[0:int(len(MovingAverage)*0.71)]
trailing_MovingAverage_test=MovingAverage[int(len(MovingAverage)*0.71):]


# Plotting Grpah for Trailing MA


plt.plot(trailing_MovingAverage_train['Sparkling'], label='Train')
plt.plot(trailing_MovingAverage_test['Sparkling'], label='Test')
plt.plot(trailing_MovingAverage_train['Trailing_2'], label='2 Point Trailing Moving Average on Training Set')
plt.plot(trailing_MovingAverage_train['Trailing_4'], label='4 Point Trailing Moving Average on Training Set')
plt.plot(trailing_MovingAverage_train['Trailing_6'],label = '6 Point Trailing Moving Average on Training Set')
plt.plot(trailing_MovingAverage_train['Trailing_9'],label = '9 Point Trailing Moving Average on Training Set')
plt.plot(trailing_MovingAverage_test['Trailing_2'], label='2 Point Trailing Moving Average on Test Set')
plt.plot(trailing_MovingAverage_test['Trailing_4'], label='4 Point Trailing Moving Average on Test Set')
plt.plot(trailing_MovingAverage_test['Trailing_6'],label = '6 Point Trailing Moving Average on Test Set')
plt.plot(trailing_MovingAverage_test['Trailing_9'],label = '9 Point Trailing Moving Average on Test Set')
plt.legend(loc = 'best')
plt.grid();


# RMSE score for Trailing MA


## Test Data - RMSE --> 2 point Trailing MA
rmse_model4_test_2 = sqrt(metrics.mean_squared_error(test['Sparkling'],trailing_MovingAverage_test['Trailing_2']))
print("For 2 point Moving Average Model forecast on the Training Data, RMSE is %3.3f" %(rmse_model4_test_2))
## Test Data - RMSE --> 4 point Trailing MA
rmse_model4_test_4 = sqrt(metrics.mean_squared_error(test['Sparkling'],trailing_MovingAverage_test['Trailing_4']))
print("For 4 point Moving Average Model forecast on the Training Data, RMSE is %3.3f" %(rmse_model4_test_4))
## Test Data - RMSE --> 6 point Trailing MA
rmse_model4_test_6 = sqrt(metrics.mean_squared_error(test['Sparkling'],trailing_MovingAverage_test['Trailing_6']))
print("For 6 point Moving Average Model forecast on the Training Data, RMSE is %3.3f" %(rmse_model4_test_6))
## Test Data - RMSE --> 9 point Trailing MA
rmse_model4_test_9 = sqrt(metrics.mean_squared_error(test['Sparkling'],trailing_MovingAverage_test['Trailing_9']))
print("For 9 point Moving Average Model forecast on the Training Data, RMSE is %3.3f " %(rmse_model4_test_9))

For 2 point Moving Average Model forecast on the Training Data, RMSE is 813.401
For 4 point Moving Average Model forecast on the Training Data, RMSE is 1156.590
For 6 point Moving Average Model forecast on the Training Data, RMSE is 1283.927
For 9 point Moving Average Model forecast on the Training Data, RMSE is 1346.278


# Creating Dataframe


resultsDf_4 = pd.DataFrame({'Test_RMSE': [rmse_model4_test_2,rmse_model4_test_4
 ,rmse_model4_test_6,rmse_model4_test_9]}
 ,index=['2pointTrailingMovingAverage','4pointTrailingMovingAverage'
 ,'6pointTrailingMovingAverage','9pointTrailingMovingAverage'])
result_DF = pd.concat([result_DF, resultsDf_4])
result_DF


## Importing Simple exponential Smoothing 
from statsmodels.tsa.holtwinters import ExponentialSmoothing, SimpleExpSmoothing, Holt


SES_Train=train.copy()
SES_test=test.copy()


model_SES=SimpleExpSmoothing(SES_Train['Sparkling'],)


model_SES_autofit=model_SES.fit(optimized=True,smoothing_level=0.995)


model_SES_autofit.params

{'smoothing_level': 0.995,
 'smoothing_trend': nan,
 'smoothing_seasonal': nan,
 'damping_trend': nan,
 'initial_level': 1686.0,
 'initial_trend': nan,
 'initial_seasons': array([], dtype=float64),
 'use_boxcox': False,
 'lamda': None,
 'remove_bias': False}


# Predicting for trainig data


SES_Train['predict'] = model_SES_autofit.fittedvalues
SES_Train.head()


# Predicting for Test data


SES_test['predict'] = model_SES_autofit.forecast(steps=55)
SES_test.head()


# Plotting graph for Simple Exponential Smooting forecast

plt.plot(SES_Train['Sparkling'], label='Train')
plt.plot(SES_test['Sparkling'], label='Test')
plt.plot(SES_Train['predict'],label='Alpha = 0.995 Simple Exponential Smoothing Predictions on Training set')
plt.plot(SES_test['predict'], label='Alpha = 0.995 Simple Exponential Smoothing predictions on Test Set')
plt.legend(loc='best')
plt.grid()
plt.title('Alpha = 0.995 Predictions');


# Train - RMSE score


rmse_model5_Train_1 = metrics.mean_squared_error(SES_Train['Sparkling'],SES_Train['predict'],squared=False)
print("For Alpha =0.995 Simple Exponential Smoothing Model forecast on the Train Data, RMSE is %3.3f" %(rmse_model5_Train_1))

For Alpha =0.995 Simple Exponential Smoothing Model forecast on the Train Data, RMSE is 1372.055


# Test - RMSE score


rmse_model5_Test_1 = metrics.mean_squared_error(SES_test['Sparkling'],SES_test['predict'],squared=False)
print("For Alpha =0.995 Simple Exponential Smoothing Model forecast on the Test Data, RMSE is %3.3f" %(rmse_model5_Test_1))

For Alpha =0.995 Simple Exponential Smoothing Model forecast on the Test Data, RMSE is 3855.941


# Creating DataFrame


resultsDf_5 = pd.DataFrame({'Train_RMSE' : [rmse_model5_Train_1],
 'Test_RMSE': [rmse_model5_Test_1]},index=['Alpha=0.995,SimpleExponentialSmoothing'])
result_DF = pd.concat([result_DF, resultsDf_5])
result_DF


# Setting Different Alpha Values
resultsDf_6 = pd.DataFrame({'Alpha_Values':[],'Train_RMSE':[],'Test_RMSE': []})
resultsDf_6


for i in np.arange(0.3,1,0.1):
 model_SES_alpha_i = model_SES.fit(smoothing_level=i,optimized=False,use_brute=True)
 SES_Train['predict',i] = model_SES_alpha_i.fittedvalues
 SES_test['predict',i] = model_SES_alpha_i.forecast(steps=55)
 
 rmse_model5_train_i = metrics.mean_squared_error(SES_Train['Sparkling'],SES_Train['predict',i],squared=False)
 
 rmse_model5_test_i = metrics.mean_squared_error(SES_test['Sparkling'],SES_test['predict',i],squared=False)
 
 resultsDf_6 = resultsDf_6.append({'Alpha_Values':i,'Train_RMSE':rmse_model5_train_i
 ,'Test_RMSE':rmse_model5_test_i}, ignore_index=True)


# Model Evaluation


resultsDf_6.sort_values(by=['Test_RMSE'],ascending=True)


# Plotting graph for different Alpha values of Traninig and Test data


plt.figure(figsize=(18,9))
plt.plot(SES_Train['Sparkling'], label='Train')
plt.plot(SES_test['Sparkling'], label='Test')
plt.plot(SES_test['predict'], label='Alpha = 1 Simple Exponential Smoothing predictions on Test Set')
plt.plot(SES_test['predict', 0.3], label='Alpha =0.3 Simple Exponential Smoothing predictions on Test Set')
plt.legend(loc='best')
plt.grid();


# Creating Dataframe


resultsDf_6_1 = pd.DataFrame({'Train_RMSE' : [resultsDf_6.sort_values(by=['Train_RMSE'],ascending=True).values[0][1]],
                              'Test_RMSE': [resultsDf_6.sort_values(by=['Test_RMSE'],ascending=True).values[0][2]]}
 ,index=['Alpha=0.3,SimpleExponentialSmoothing'])
result_DF = pd.concat([result_DF, resultsDf_6_1])
result_DF


# Model 6 : Double Exponential Smoothing

DES_train = train.copy()
DES_test = test.copy()


model_DES = Holt(DES_train['Sparkling'])


## First we will define an empty dataframe to store our values from the loop
resultsDf_7 = pd.DataFrame({'Alpha_Values':[],'Beta_Values':[],'Train_RMSE':[],'Test_RMSE': []})
resultsDf_7


for i in np.arange(0.3,1.1,0.1):
    for j in np.arange(0.3,1.1,0.1):
        model_DES_alpha_i_j = model_DES.fit(smoothing_level=i,smoothing_trend=j,optimized=False,use_brute=True)
        DES_train['predict',i,j] = model_DES_alpha_i_j.fittedvalues
        DES_test['predict',i,j] = model_DES_alpha_i_j.forecast(steps=55)
        
        rmse_model6_train = metrics.mean_squared_error(DES_train['Sparkling'],DES_train['predict',i,j],squared=False)
        
        rmse_model6_test = metrics.mean_squared_error(DES_test['Sparkling'],DES_test['predict',i,j],squared=False)
        
        resultsDf_7 = resultsDf_7.append({'Alpha_Values':i,'Beta_Values':j,'Train_RMSE':rmse_model6_train,'Test_RMSE':rmse_model6_test}, ignore_index=True)


resultsDf_7


resultsDf_7.sort_values(by=['Test_RMSE']).head()


# Plotting Graph on Both Traning and Test data


plt.figure(figsize=(18,9))
plt.plot(DES_train['Sparkling'], label='Train')
plt.plot(DES_test['Sparkling'], label='Test')
plt.plot(DES_train['predict', 0.3, 0.3], label='Alpha=0.3,Beta=0.3,DoubleExponentialSmoothing predictions on Training set')
plt.plot(DES_test['predict', 0.3, 0.3], label='Alpha=0.3,Beta=0.3,DoubleExponentialSmoothing predictions on Test Set')
plt.legend(loc='best')
plt.grid();


resultsDf_7_1 = pd.DataFrame({'Train_RMSE' :[resultsDf_7.sort_values(by=['Train_RMSE']).values[0][2]],
 'Test_RMSE': [resultsDf_7.sort_values(by=['Test_RMSE']).values[0][3]]}
 ,index=['Alpha=0.3,Beta=0.3,DoubleExponentialSmoothing'])
result_DF = pd.concat([result_DF, resultsDf_7_1])
result_DF


TES_train = train.copy()
TES_test = test.copy()


model_TES = ExponentialSmoothing(TES_train['Sparkling'],trend='additive',seasonal='multiplicative',freq='M')


model_TES_autofit = model_TES.fit()


model_TES_autofit.params

{'smoothing_level': 0.11057044018305404,
 'smoothing_trend': 0.06076609768412894,
 'smoothing_seasonal': 0.39187601902826213,
 'damping_trend': nan,
 'initial_level': 1621.806699459997,
 'initial_trend': -8.175193580026457,
 'initial_seasons': array([1.07304448, 1.02730389, 1.39469706, 1.20333193, 0.98008967,
        0.97664181, 1.39261648, 1.70888984, 1.37519684, 1.81953245,
        2.82880203, 3.59142504]),
 'use_boxcox': False,
 'lamda': None,
 'remove_bias': False}


# Predicton on Train Set


TES_train['auto_predict'] = model_TES_autofit.fittedvalues
TES_train.head()


# Prediction on Test set


TES_test['auto_predict'] = model_TES_autofit.forecast(steps=len(test))
TES_test.head()


# Plotting graph using Training set and Test set using autofit


plt.plot(TES_train['Sparkling'], label='Train')
plt.plot(TES_test['Sparkling'], label='Test')
plt.plot(TES_train['auto_predict'], label='Alpha=0.064,Beta=0.053,Gamma=0.0,TripleExponentialSmoothing predictions on Test Set')
plt.plot(TES_test['auto_predict'], label='Alpha=0.064,Beta=0.053,Gamma=0.0,TripleExponentialSmoothing predictions on Test Set') 
plt.legend(loc='best')
plt.grid();


# Train - RMSE score


rmse_model6_train_1 = metrics.mean_squared_error(TES_train['Sparkling'],TES_train['auto_predict'],squared=False)
print("For Alpha=0.064,Beta=0.053,Gamma=0.0, Triple Exponential Smoothing Model forecast on the Train Data, RMSE is %3.3f" %(rmse_model6_train_1))

For Alpha=0.064,Beta=0.053,Gamma=0.0, Triple Exponential Smoothing Model forecast on the Train Data, RMSE is 356.782


rmse_model6_test_1 = metrics.mean_squared_error(TES_test['Sparkling'],TES_test['auto_predict'],squared=False)
print("For Alpha=0.064,Beta=0.053,Gamma=0.0, Triple Exponential Smoothing Model forecast on the Test Data, RMSE is %3.3f" %(rmse_model6_test_1))

For Alpha=0.064,Beta=0.053,Gamma=0.0, Triple Exponential Smoothing Model forecast on the Test Data, RMSE is 463.502


# Creating DataFrame


resultsDf_8_1 = pd.DataFrame({'Train_RMSE':[rmse_model6_train_1],
 'Test_RMSE': [rmse_model6_test_1]}
 ,index=['Alpha=0.064,Beta=0.053,Gamma=0.0,TripleExponentialSmoothing'])
result_DF = pd.concat([result_DF, resultsDf_8_1])
result_DF


## First we will define an empty dataframe to store our values from the loop
resultsDf_8_2 = pd.DataFrame({'Alpha_Values':[],'Beta_Values':[],'Gamma_Values':[],'Train_RMSE':[],'Test_RMSE': []})
resultsDf_8_2


for i in np.arange(0.3,1.1,0.1):
    for j in np.arange(0.3,1.1,0.1):
         for k in np.arange(0.3,1.1,0.1):
                model_TES_alpha_i_j_k = model_TES.fit(smoothing_level=i,smoothing_trend=j,smoothing_seasonal=k,optimized=False,use_brute=True)
                TES_train['predict',i,j,k] = model_TES_alpha_i_j_k.fittedvalues
                TES_test['predict',i,j,k] = model_TES_alpha_i_j_k.forecast(steps=55)
                rmse_model8_train = metrics.mean_squared_error(TES_train['Sparkling'],TES_train['predict',i,j,k],squared=False)
                rmse_model8_test = metrics.mean_squared_error(TES_test['Sparkling'],TES_test['predict',i,j,k],squared=False)
                resultsDf_8_2 = resultsDf_8_2.append({'Alpha_Values':i,'Beta_Values':j,'Gamma_Values':k,
                                                          'Train_RMSE':rmse_model8_train,'Test_RMSE':rmse_model8_test}
                                                          , ignore_index=True)


resultsDf_8_2


resultsDf_8_2.sort_values(by=['Test_RMSE']).head()


## Plotting on both the Training and Test data using brute force alpha, beta and gamma determination
plt.plot(TES_train['Sparkling'], label='Train')
plt.plot(TES_test['Sparkling'], label='Test')
#The value of alpha and beta is taken like that by python
plt.plot(TES_train['predict', 0.3, 0.4, 0.3], label='Alpha=0.3,Beta=0.4,Gamma=0.3,TripleExponentialSmoothing prediction on Training set')
plt.plot(TES_test['predict', 0.3, 0.4, 0.3], label='Alpha=0.3,Beta=0.4,Gamma=0.3,TripleExponentialSmoothing prediction on Testing set')
plt.legend(loc='best')
plt.grid();


# Creating Dataframe


resultsDf_8_3 = pd.DataFrame({'Train_RMSE':[resultsDf_8_2.sort_values(by=['Test_RMSE']).values[0][3]],
 'Test_RMSE': [resultsDf_8_2.sort_values(by=['Test_RMSE']).values[0][4]]}
 ,index=['Alpha=0.3,Beta=0.4,Gamma=0.3,TripleExponentialSmoothing'])
result_DF = pd.concat([result_DF, resultsDf_8_3])
result_DF


print('Sorted by RMSE values on Test Data','\n',result_DF.sort_values(by=['Test_RMSE']),'\n','\n')
print('Sorted by RMSE values on Train Data','\n',result_DF.sort_values(by=['Train_RMSE']))

Sorted by RMSE values on Test Data 
                                                      Train_RMSE     Test_RMSE
Alpha=0.3,Beta=0.4,Gamma=0.3,TripleExponentialS...   404.513320    392.786198
Alpha=0.064,Beta=0.053,Gamma=0.0,TripleExponent...   356.782453    463.501976
2pointTrailingMovingAverage                                 NaN    813.400684
4pointTrailingMovingAverage                                 NaN   1156.589694
Simple Average                                      1298.483628   1275.081804
6pointTrailingMovingAverage                                 NaN   1283.927428
9pointTrailingMovingAverage                                 NaN   1346.278315
RegressionOnTime                                    1279.322346   1389.135175
Alpha=0.3,SimpleExponentialSmoothing                1338.805381   1935.507132
Alpha=0.995,SimpleExponentialSmoothing              1372.054747   3855.940897
Naive Model                                         3867.700802   3864.279352
Alpha=0.3,Beta=0.3,DoubleExponentialSmoothing       1500.689062  18259.110704 
 

Sorted by RMSE values on Train Data 
                                                      Train_RMSE     Test_RMSE
Alpha=0.064,Beta=0.053,Gamma=0.0,TripleExponent...   356.782453    463.501976
Alpha=0.3,Beta=0.4,Gamma=0.3,TripleExponentialS...   404.513320    392.786198
RegressionOnTime                                    1279.322346   1389.135175
Simple Average                                      1298.483628   1275.081804
Alpha=0.3,SimpleExponentialSmoothing                1338.805381   1935.507132
Alpha=0.995,SimpleExponentialSmoothing              1372.054747   3855.940897
Alpha=0.3,Beta=0.3,DoubleExponentialSmoothing       1500.689062  18259.110704
Naive Model                                         3867.700802   3864.279352
2pointTrailingMovingAverage                                 NaN    813.400684
4pointTrailingMovingAverage                                 NaN   1156.589694
6pointTrailingMovingAverage                                 NaN   1283.927428
9pointTrailingMovingAverage                                 NaN   1346.278315


from statsmodels.tsa.stattools import adfuller
def test_stationarity(timeseries):
    rolmean = timeseries.rolling(window=7).mean() #determining the rolling mean
    rolstd = timeseries.rolling(window=7).std() #determining the rolling standard deviation
    orig = plt.plot(timeseries, color='blue',label='Original')#Plot rolling statistics:
    mean = plt.plot(rolmean, color='red', label='Rolling Mean')
    std = plt.plot(rolstd, color='black', label = 'Rolling Std')
    plt.legend(loc='best')
    plt.title('Rolling Mean & Standard Deviation')
    plt.show(block=False)
    
    #Perform Dickey-Fuller test:
    print ('Results of Dickey-Fuller Test:')
    dftest = adfuller(timeseries, autolag='AIC')
    dfoutput = pd.Series(dftest[0:4], index=['Test Statistic','p-value','#Lags Used','Number of Observations Used'])
    
    for key,value in dftest[4].items():
        dfoutput['Critical Value (%s)'%key] = value
        print (dfoutput,'\n')


test_stationarity(df['Sparkling'])

Results of Dickey-Fuller Test:
Test Statistic                  -1.360497
p-value                          0.601061
#Lags Used                      11.000000
Number of Observations Used    175.000000
Critical Value (1%)             -3.468280
dtype: float64 

Test Statistic                  -1.360497
p-value                          0.601061
#Lags Used                      11.000000
Number of Observations Used    175.000000
Critical Value (1%)             -3.468280
Critical Value (5%)             -2.878202
dtype: float64 

Test Statistic                  -1.360497
p-value                          0.601061
#Lags Used                      11.000000
Number of Observations Used    175.000000
Critical Value (1%)             -3.468280
Critical Value (5%)             -2.878202
Critical Value (10%)            -2.575653
dtype: float64


# Let us take a difference of order 1 and check whether the Time Series is stationary or not.


test_stationarity(df['Sparkling'].diff().dropna())

Results of Dickey-Fuller Test:
Test Statistic                 -45.050301
p-value                          0.000000
#Lags Used                      10.000000
Number of Observations Used    175.000000
Critical Value (1%)             -3.468280
dtype: float64 

Test Statistic                 -45.050301
p-value                          0.000000
#Lags Used                      10.000000
Number of Observations Used    175.000000
Critical Value (1%)             -3.468280
Critical Value (5%)             -2.878202
dtype: float64 

Test Statistic                 -45.050301
p-value                          0.000000
#Lags Used                      10.000000
Number of Observations Used    175.000000
Critical Value (1%)             -3.468280
Critical Value (5%)             -2.878202
Critical Value (10%)            -2.575653
dtype: float64


## The following loop helps us in getting a combination of different parameters of p and q in the range of 0 and 2
## We have kept the value of d as 1 as we need to take a difference of the series to make it stationary.
import itertools
p = q = range(0, 3)
d= range(1,2)
pdq = list(itertools.product(p, d, q))
print('Some parameter combinations for the Model...')
for i in range(1,len(pdq)):
    print('Model: {}'.format(pdq[i]))

Some parameter combinations for the Model...
Model: (0, 1, 1)
Model: (0, 1, 2)
Model: (1, 1, 0)
Model: (1, 1, 1)
Model: (1, 1, 2)
Model: (2, 1, 0)
Model: (2, 1, 1)
Model: (2, 1, 2)


# Creating an empty Dataframe with column names only
ARIMA_AIC = pd.DataFrame(columns=['param', 'AIC'])
ARIMA_AIC


from statsmodels.tsa.arima_model import ARIMA
import warnings
warnings.filterwarnings('ignore')
for param in pdq:
    ARIMA_model = ARIMA(train['Sparkling'].values,order=param).fit()
    print('ARIMA{} - AIC:{}'.format(param,ARIMA_model.aic))
    ARIMA_AIC = ARIMA_AIC.append({'param':param, 'AIC': ARIMA_model.aic}, ignore_index=True)

ARIMA(0, 1, 0) - AIC:2269.582796371201
ARIMA(0, 1, 1) - AIC:2264.9064368172944
ARIMA(0, 1, 2) - AIC:2232.783097684661
ARIMA(1, 1, 0) - AIC:2268.5280606648653
ARIMA(1, 1, 1) - AIC:2235.0139453501533
ARIMA(1, 1, 2) - AIC:2233.597647119118
ARIMA(2, 1, 0) - AIC:2262.035600095461
ARIMA(2, 1, 1) - AIC:2232.3604898819262
ARIMA(2, 1, 2) - AIC:2210.6185075458225


## Sort the above AIC values in the ascending order to get the parameters for the minimum AIC value
ARIMA_AIC.sort_values(by='AIC',ascending=True)


auto_ARIMA = ARIMA(train['Sparkling'], order=(0,1,2),freq='M')
results_auto_ARIMA = auto_ARIMA.fit()
print(results_auto_ARIMA.summary())

                             ARIMA Model Results                              
==============================================================================
Dep. Variable:            D.Sparkling   No. Observations:                  131
Model:                 ARIMA(0, 1, 2)   Log Likelihood               -1112.392
Method:                       css-mle   S.D. of innovations           1159.696
Date:                Thu, 28 Oct 2021   AIC                           2232.783
Time:                        19:14:16   BIC                           2244.284
Sample:                    02-29-1980   HQIC                          2237.456
                         - 12-31-1990                                         
=====================================================================================
                        coef    std err          z      P>|z|      [0.025      0.975]
-------------------------------------------------------------------------------------
const                 6.2472      3.800      1.644      0.100      -1.201      13.696
ma.L1.D.Sparkling    -0.5555      0.073     -7.583      0.000      -0.699      -0.412
ma.L2.D.Sparkling    -0.4445      0.071     -6.247      0.000      -0.584      -0.305
                                    Roots                                    
=============================================================================
                  Real          Imaginary           Modulus         Frequency
-----------------------------------------------------------------------------
MA.1            1.0000           +0.0000j            1.0000            0.0000
MA.2           -2.2495           +0.0000j            2.2495            0.5000
-----------------------------------------------------------------------------


# Predict on the Test Set using this model and evaluate the model.


predicted_auto_ARIMA = results_auto_ARIMA.forecast(steps=len(test))


from sklearn.metrics import mean_squared_error
rmse = mean_squared_error(test['Sparkling'],predicted_auto_ARIMA[0],squared=False)
print(rmse)

1417.502239430773


results_DF= pd.DataFrame({'RMSE': [rmse]}
 ,index=['ARIMA(0,1,2)'])
results_DF


# Build an Automated version of a SARIMA model for which the best parameters are selected in accordance with the lowest Akaike Information Criteria (AIC)


import itertools
p = q = range(0, 3)
d= range(1,2)
D = range(0,1)
pdq = list(itertools.product(p, d, q))
model_pdq = [(x[0], x[1], x[2], 12) for x in list(itertools.product(p, D, q))]
print('Examples of some parameter combinations for Model...')
for i in range(1,len(pdq)):
    print('Model: {}{}'.format(pdq[i], model_pdq[i]))

Examples of some parameter combinations for Model...
Model: (0, 1, 1)(0, 0, 1, 12)
Model: (0, 1, 2)(0, 0, 2, 12)
Model: (1, 1, 0)(1, 0, 0, 12)
Model: (1, 1, 1)(1, 0, 1, 12)
Model: (1, 1, 2)(1, 0, 2, 12)
Model: (2, 1, 0)(2, 0, 0, 12)
Model: (2, 1, 1)(2, 0, 1, 12)
Model: (2, 1, 2)(2, 0, 2, 12)


SARIMA_AIC = pd.DataFrame(columns=['param','seasonal', 'AIC'])
SARIMA_AIC


import statsmodels.api as sm

for param in pdq:
    for param_seasonal in model_pdq:
        SARIMA_model = sm.tsa.statespace.SARIMAX(train['Sparkling'].values,
                                                 order=param,
                                                 seasonal_order=param_seasonal,
                                                 enforce_stationarity=False,
                                                 enforce_invertibility=False)
        results_SARIMA = SARIMA_model.fit(maxiter=1000)
        print('SARIMA{}x{} - AIC:{}'.format(param, param_seasonal, results_SARIMA.aic))
        SARIMA_AIC = SARIMA_AIC.append({'param':param,'seasonal':param_seasonal ,'AIC': results_SARIMA.aic}, ignore_index=True)

SARIMA(0, 1, 0)x(0, 0, 0, 12) - AIC:2251.3597196862966
SARIMA(0, 1, 0)x(0, 0, 1, 12) - AIC:1956.261461684684
SARIMA(0, 1, 0)x(0, 0, 2, 12) - AIC:1723.153364023882
SARIMA(0, 1, 0)x(1, 0, 0, 12) - AIC:1837.4366022456675
SARIMA(0, 1, 0)x(1, 0, 1, 12) - AIC:1806.9905301389067
SARIMA(0, 1, 0)x(1, 0, 2, 12) - AIC:1633.2108735791512
SARIMA(0, 1, 0)x(2, 0, 0, 12) - AIC:1648.3776153470856
SARIMA(0, 1, 0)x(2, 0, 1, 12) - AIC:1647.2054158617952
SARIMA(0, 1, 0)x(2, 0, 2, 12) - AIC:1630.989805392081
SARIMA(0, 1, 1)x(0, 0, 0, 12) - AIC:2230.162907850583
SARIMA(0, 1, 1)x(0, 0, 1, 12) - AIC:1923.768864956663
SARIMA(0, 1, 1)x(0, 0, 2, 12) - AIC:1692.708957284665
SARIMA(0, 1, 1)x(1, 0, 0, 12) - AIC:1797.1795881838295
SARIMA(0, 1, 1)x(1, 0, 1, 12) - AIC:1738.090319377895
SARIMA(0, 1, 1)x(1, 0, 2, 12) - AIC:1570.1319656885905
SARIMA(0, 1, 1)x(2, 0, 0, 12) - AIC:1605.6751954175527
SARIMA(0, 1, 1)x(2, 0, 1, 12) - AIC:1599.2245085468476
SARIMA(0, 1, 1)x(2, 0, 2, 12) - AIC:1570.4018829870888
SARIMA(0, 1, 2)x(0, 0, 0, 12) - AIC:2187.4410101687313
SARIMA(0, 1, 2)x(0, 0, 1, 12) - AIC:1887.9128007201634
SARIMA(0, 1, 2)x(0, 0, 2, 12) - AIC:1659.8789890562743
SARIMA(0, 1, 2)x(1, 0, 0, 12) - AIC:1790.0326332151465
SARIMA(0, 1, 2)x(1, 0, 1, 12) - AIC:1724.167507097238
SARIMA(0, 1, 2)x(1, 0, 2, 12) - AIC:1557.160506811068
SARIMA(0, 1, 2)x(2, 0, 0, 12) - AIC:1603.9654774510207
SARIMA(0, 1, 2)x(2, 0, 1, 12) - AIC:1600.543880243975
SARIMA(0, 1, 2)x(2, 0, 2, 12) - AIC:1557.1215630847455
SARIMA(1, 1, 0)x(0, 0, 0, 12) - AIC:2250.3181267386713
SARIMA(1, 1, 0)x(0, 0, 1, 12) - AIC:1954.3938339903953
SARIMA(1, 1, 0)x(0, 0, 2, 12) - AIC:1721.2688476354715
SARIMA(1, 1, 0)x(1, 0, 0, 12) - AIC:1811.244027933166
SARIMA(1, 1, 0)x(1, 0, 1, 12) - AIC:1788.5343594187407
SARIMA(1, 1, 0)x(1, 0, 2, 12) - AIC:1616.4894402576397
SARIMA(1, 1, 0)x(2, 0, 0, 12) - AIC:1621.6355080128965
SARIMA(1, 1, 0)x(2, 0, 1, 12) - AIC:1617.1356132769517
SARIMA(1, 1, 0)x(2, 0, 2, 12) - AIC:1616.5412067564307
SARIMA(1, 1, 1)x(0, 0, 0, 12) - AIC:2204.9340491545727
SARIMA(1, 1, 1)x(0, 0, 1, 12) - AIC:1907.355897413186
SARIMA(1, 1, 1)x(0, 0, 2, 12) - AIC:1678.0981352612985
SARIMA(1, 1, 1)x(1, 0, 0, 12) - AIC:1775.1424469157992
SARIMA(1, 1, 1)x(1, 0, 1, 12) - AIC:1739.716746888626
SARIMA(1, 1, 1)x(1, 0, 2, 12) - AIC:1571.3248863617944
SARIMA(1, 1, 1)x(2, 0, 0, 12) - AIC:1590.616160688121
SARIMA(1, 1, 1)x(2, 0, 1, 12) - AIC:1586.3140616479732
SARIMA(1, 1, 1)x(2, 0, 2, 12) - AIC:1571.8069972326896
SARIMA(1, 1, 2)x(0, 0, 0, 12) - AIC:2188.46334505047
SARIMA(1, 1, 2)x(0, 0, 1, 12) - AIC:1889.7708307500288
SARIMA(1, 1, 2)x(0, 0, 2, 12) - AIC:1659.6291421470712
SARIMA(1, 1, 2)x(1, 0, 0, 12) - AIC:1771.8259798997447
SARIMA(1, 1, 2)x(1, 0, 1, 12) - AIC:1723.9952183667806
SARIMA(1, 1, 2)x(1, 0, 2, 12) - AIC:1555.5842472256138
SARIMA(1, 1, 2)x(2, 0, 0, 12) - AIC:1588.4214728481102
SARIMA(1, 1, 2)x(2, 0, 1, 12) - AIC:1585.5250820812246
SARIMA(1, 1, 2)x(2, 0, 2, 12) - AIC:1556.0767703468919
SARIMA(2, 1, 0)x(0, 0, 0, 12) - AIC:2227.302761872421
SARIMA(2, 1, 0)x(0, 0, 1, 12) - AIC:1946.4383435408224
SARIMA(2, 1, 0)x(0, 0, 2, 12) - AIC:1711.4123039832598
SARIMA(2, 1, 0)x(1, 0, 0, 12) - AIC:1780.764606605774
SARIMA(2, 1, 0)x(1, 0, 1, 12) - AIC:1756.9357366284632
SARIMA(2, 1, 0)x(1, 0, 2, 12) - AIC:1600.970220427088
SARIMA(2, 1, 0)x(2, 0, 0, 12) - AIC:1592.240346481058
SARIMA(2, 1, 0)x(2, 0, 1, 12) - AIC:1587.6344990137136
SARIMA(2, 1, 0)x(2, 0, 2, 12) - AIC:1585.9191733172352
SARIMA(2, 1, 1)x(0, 0, 0, 12) - AIC:2199.8586131454495
SARIMA(2, 1, 1)x(0, 0, 1, 12) - AIC:1905.0209495915058
SARIMA(2, 1, 1)x(0, 0, 2, 12) - AIC:1675.423408032731
SARIMA(2, 1, 1)x(1, 0, 0, 12) - AIC:1792.823429063149
SARIMA(2, 1, 1)x(1, 0, 1, 12) - AIC:1740.0911252155481
SARIMA(2, 1, 1)x(1, 0, 2, 12) - AIC:1571.9888279741326
SARIMA(2, 1, 1)x(2, 0, 0, 12) - AIC:1577.1235060896936
SARIMA(2, 1, 1)x(2, 0, 1, 12) - AIC:1573.159584985324
SARIMA(2, 1, 1)x(2, 0, 2, 12) - AIC:1572.3428636863346
SARIMA(2, 1, 2)x(0, 0, 0, 12) - AIC:2176.8681150587977
SARIMA(2, 1, 2)x(0, 0, 1, 12) - AIC:1892.237264771439
SARIMA(2, 1, 2)x(0, 0, 2, 12) - AIC:1661.5523433555347
SARIMA(2, 1, 2)x(1, 0, 0, 12) - AIC:1757.2140931305535
SARIMA(2, 1, 2)x(1, 0, 1, 12) - AIC:1725.608605288059
SARIMA(2, 1, 2)x(1, 0, 2, 12) - AIC:1557.3404092912224
SARIMA(2, 1, 2)x(2, 0, 0, 12) - AIC:1599.7923667501423
SARIMA(2, 1, 2)x(2, 0, 1, 12) - AIC:1573.547601420174
SARIMA(2, 1, 2)x(2, 0, 2, 12) - AIC:1557.840126824272


SARIMA_AIC.sort_values(by=['AIC']).head()


import statsmodels.api as sm

auto_SARIMA_12 = sm.tsa.statespace.SARIMAX(train['Sparkling'].values,
                                         order=(0, 1, 2),
                                         seasonal_order=(2, 0, 2, 12),
                                         enforce_stationarity=False,
                                         enforce_invertibility=False)
 
results_auto_SARIMA_12 = auto_SARIMA_12.fit(maxiter=1000)
print(results_auto_SARIMA_12.summary())

                                     SARIMAX Results                                      
==========================================================================================
Dep. Variable:                                  y   No. Observations:                  132
Model:             SARIMAX(0, 1, 2)x(2, 0, 2, 12)   Log Likelihood                -771.561
Date:                            Thu, 28 Oct 2021   AIC                           1557.122
Time:                                    19:16:05   BIC                           1575.632
Sample:                                         0   HQIC                          1564.621
                                            - 132                                         
Covariance Type:                              opg                                         
==============================================================================
                 coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------
ma.L1         -0.7771      0.101     -7.680      0.000      -0.975      -0.579
ma.L2         -0.1239      0.121     -1.022      0.307      -0.361       0.114
ar.S.L12       0.6982      0.751      0.930      0.353      -0.774       2.170
ar.S.L24       0.3598      0.783      0.460      0.646      -1.174       1.894
ma.S.L12       1.6415      3.715      0.442      0.659      -5.640       8.923
ma.S.L24      -1.5933      2.663     -0.598      0.550      -6.813       3.627
sigma2      2.915e+04   9.51e+04      0.307      0.759   -1.57e+05    2.15e+05
===================================================================================
Ljung-Box (L1) (Q):                   0.03   Jarque-Bera (JB):                22.07
Prob(Q):                              0.85   Prob(JB):                         0.00
Heteroskedasticity (H):               1.45   Skew:                             0.49
Prob(H) (two-sided):                  0.28   Kurtosis:                         5.04
===================================================================================

Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).


results_auto_SARIMA_12.plot_diagnostics()
plt.show()


# Predict on the Test Set using this model and evaluate the model.


predicted_auto_SARIMA_12 = results_auto_SARIMA_12.get_forecast(steps=len(test))


predicted_auto_SARIMA_12.summary_frame(alpha=0.05).head()


rmse = mean_squared_error(test['Sparkling'],predicted_auto_SARIMA_12.predicted_mean,squared=False)
print(rmse)

526.4628785794362


temp_resultsDf = pd.DataFrame({'RMSE': [rmse]}
 ,index=['SARIMA(0,1,2)(2,0,2,12)'])
results_DF = pd.concat([results_DF,temp_resultsDf])
results_DF


from statsmodels.graphics.tsaplots import plot_acf, plot_pacf
plot_acf(df['Sparkling'].diff().dropna(),lags=50,title='Differenced Data Autocorrelation')
plot_pacf(df['Sparkling'].diff().dropna(),lags=50,title='Differenced Data Patial Autocorrelation')
plt.show()


df.plot()
plt.grid();


(df['Sparkling'].diff(12)).plot()
plt.grid();


(df['Sparkling'].diff(12)).diff().plot()
plt.grid();


# Let us go ahead and check the stationarity of the above series before fitting the SARIMA model.
test_stationarity((train['Sparkling'].diff(12).dropna()).diff(1).dropna())

Results of Dickey-Fuller Test:
Test Statistic                  -3.342905
p-value                          0.013066
#Lags Used                      10.000000
Number of Observations Used    108.000000
Critical Value (1%)             -3.492401
dtype: float64 

Test Statistic                  -3.342905
p-value                          0.013066
#Lags Used                      10.000000
Number of Observations Used    108.000000
Critical Value (1%)             -3.492401
Critical Value (5%)             -2.888697
dtype: float64 

Test Statistic                  -3.342905
p-value                          0.013066
#Lags Used                      10.000000
Number of Observations Used    108.000000
Critical Value (1%)             -3.492401
Critical Value (5%)             -2.888697
Critical Value (10%)            -2.581255
dtype: float64


# Checking the ACF and the PACF plots for the new modified Time Series
plot_acf((df['Sparkling'].diff(12).dropna()).diff(1).dropna(),lags=30)
plot_pacf((df['Sparkling'].diff(12).dropna()).diff(1).dropna(),lags=30);


# We are going to take the seasonal period as 12.


manual_SARIMA_12= sm.tsa.statespace.SARIMAX(train['Sparkling'].values,
 order=(2, 1, 2),
 seasonal_order=(2, 0, 2, 12),
 enforce_stationarity=False,
 enforce_invertibility=False)
results_manual_SARIMA_12 = manual_SARIMA_12.fit(maxiter=1000)
print(results_manual_SARIMA_12.summary())

                                     SARIMAX Results                                      
==========================================================================================
Dep. Variable:                                  y   No. Observations:                  132
Model:             SARIMAX(2, 1, 2)x(2, 0, 2, 12)   Log Likelihood                -769.920
Date:                            Thu, 28 Oct 2021   AIC                           1557.840
Time:                                    19:17:04   BIC                           1581.640
Sample:                                         0   HQIC                          1567.482
                                            - 132                                         
Covariance Type:                              opg                                         
==============================================================================
                 coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------
ar.L1         -0.6602      0.287     -2.300      0.021      -1.223      -0.098
ar.L2         -0.0621      0.155     -0.399      0.690      -0.367       0.243
ma.L1         -0.1042      0.272     -0.383      0.702      -0.638       0.429
ma.L2         -0.6909      0.230     -2.999      0.003      -1.142      -0.239
ar.S.L12       0.7516      0.593      1.268      0.205      -0.410       1.913
ar.S.L24       0.3072      0.617      0.498      0.618      -0.902       1.516
ma.S.L12      -0.3174      0.585     -0.542      0.588      -1.464       0.829
ma.S.L24      -0.2745      0.370     -0.742      0.458      -1.000       0.451
sigma2      1.519e+05   2.08e+04      7.321      0.000    1.11e+05    1.93e+05
===================================================================================
Ljung-Box (L1) (Q):                   0.00   Jarque-Bera (JB):                10.63
Prob(Q):                              0.98   Prob(JB):                         0.00
Heteroskedasticity (H):               1.61   Skew:                             0.29
Prob(H) (two-sided):                  0.17   Kurtosis:                         4.45
===================================================================================

Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).


results_manual_SARIMA_12.plot_diagnostics()
plt.show()


# Predict on the Test Set using this model and evaluate the model.


predicted_manual_SARIMA_12 = results_manual_SARIMA_12.get_forecast(steps=len(test))


predicted_manual_SARIMA_12.summary_frame(alpha=0.05).head()


# RMSE Score
rmse = mean_squared_error(test['Sparkling'],predicted_manual_SARIMA_12.predicted_mean,squared=False)
print(rmse)

580.6029005672854


temp_resultsDf = pd.DataFrame({'RMSE': [rmse]}
 ,index=['SARIMA(2,1,2)(2,0,2,12)'])
results_DF = pd.concat([results_DF,temp_resultsDf])
results_DF


# Manual ARIMA model
###Let us look once agin ACF and PACF plots
plot_acf(df['Sparkling'].diff().dropna(),lags=50,title='Differenced Data Autocorrelation')
plot_pacf(df['Sparkling'].diff().dropna(),lags=50,title='Differenced Data Partial Autocorrelation')
plt.show()


manual_ARIMA = ARIMA(train['Sparkling'].astype('float64'), order=(2,1,2),freq='M')
results_manual_ARIMA = manual_ARIMA.fit()
print(results_manual_ARIMA.summary())

                             ARIMA Model Results                              
==============================================================================
Dep. Variable:            D.Sparkling   No. Observations:                  131
Model:                 ARIMA(2, 1, 2)   Log Likelihood               -1099.309
Method:                       css-mle   S.D. of innovations           1012.629
Date:                Thu, 28 Oct 2021   AIC                           2210.619
Time:                        19:17:52   BIC                           2227.870
Sample:                    02-29-1980   HQIC                          2217.628
                         - 12-31-1990                                         
=====================================================================================
                        coef    std err          z      P>|z|      [0.025      0.975]
-------------------------------------------------------------------------------------
const                 5.5856      0.517     10.809      0.000       4.573       6.598
ar.L1.D.Sparkling     1.2699      0.074     17.046      0.000       1.124       1.416
ar.L2.D.Sparkling    -0.5601      0.074     -7.617      0.000      -0.704      -0.416
ma.L1.D.Sparkling    -1.9980      0.042    -47.159      0.000      -2.081      -1.915
ma.L2.D.Sparkling     0.9980      0.042     23.542      0.000       0.915       1.081
                                    Roots                                    
=============================================================================
                  Real          Imaginary           Modulus         Frequency
-----------------------------------------------------------------------------
AR.1            1.1335           -0.7073j            1.3361           -0.0888
AR.2            1.1335           +0.7073j            1.3361            0.0888
MA.1            1.0002           +0.0000j            1.0002            0.0000
MA.2            1.0019           +0.0000j            1.0019            0.0000
-----------------------------------------------------------------------------


# Predict on the Test Set using this model and evaluate the model
predicted_manual_ARIMA = results_manual_ARIMA.forecast(steps=len(test))


rmse = mean_squared_error(test['Sparkling'],predicted_manual_ARIMA[0],squared=False)
print(rmse)

1374.6638702744171


temp_resultsDf = pd.DataFrame({'RMSE': [rmse]}
 ,index=['ARIMA(2,1,2)'])
results_DF = pd.concat([results_DF,temp_resultsDf])
results_DF


results_DF = results_DF.rename(columns={'RMSE':'Test_RMSE'})


results_DF


Final_Model_results=pd.concat([result_DF,results_DF])
Final_Model_results


Final_Model_results.sort_values(by=['Test_RMSE'],ascending=True)


TES_Final=df.copy()


model_TES=ExponentialSmoothing(TES_Final['Sparkling'],trend='additive',seasonal='multiplicative',freq='M')


model_TES_autofit = model_TES.fit()


model_TES_autofit.params

{'smoothing_level': 0.07543819037108564,
 'smoothing_trend': 0.07543816249429056,
 'smoothing_seasonal': 0.27294445154459185,
 'damping_trend': nan,
 'initial_level': 1577.446620339965,
 'initial_trend': -13.373823757745805,
 'initial_seasons': array([1.06875929, 1.02018982, 1.47799192, 1.19853279, 0.98227378,
        0.9560861 , 1.32099751, 1.70927388, 1.39014226, 1.89705741,
        2.90535421, 3.77843847]),
 'use_boxcox': False,
 'lamda': None,
 'remove_bias': False}


TES_Final['auto']=model_TES_autofit.fittedvalues
TES_Final


TES_Future= model_TES_autofit.forecast(steps=12)
TES_Future

1995-08-31    1929.565002
1995-09-30    2347.047048
1995-10-31    3172.507937
1995-11-30    3909.340833
1995-12-31    5969.868543
1996-01-31    1353.953302
1996-02-29    1593.630283
1996-03-31    1824.817614
1996-04-30    1784.274585
1996-05-31    1635.069228
1996-06-30    1549.085128
1996-07-31    1955.711909
Freq: M, dtype: float64


# Plotting The Forecast with Confidence Bond
plt.figure(figsize=(18,9))
plt.plot(TES_Final,label='Current')
plt.plot(TES_Future,label='Future')
plt.plot(TES_Future, label=['Alpha=0.107,Beta=1.38,Gamma=0.0002,TripleExponentialSmoothing prediction in future for 12 Months'])
plt.legend(loc='best')
plt.grid();


# Final Result
rmse= metrics.mean_squared_error(TES_Final['Sparkling'],TES_Final['auto'],squared=False)
print('For Alpha=0.107,Beta=1.38,Gamma=0.0002 , Triple exponential Smoothing model Forecasting',rmse)

For Alpha=0.107,Beta=1.38,Gamma=0.0002 , Triple exponential Smoothing model Forecasting 346.1249703143661


# Predicted Model for the Sparkling Sales for final Model
TES_Final['Sparkling'].plot()
TES_Future.plot()

<AxesSubplot:xlabel='Time_Stamp'>


TES_Final_1=pd.DataFrame({'lower_CI':TES_Future - ((np.percentile(TES_Future,[2.5,97.5])))[0],
                         'Forecasting': TES_Future,
                         'upper_CI': TES_Future + ((np.percentile(TES_Future,[2.5,97.5])))[1]})


axis=df['Sparkling'].plot(label='Actual',figsize=(15,8))
TES_Final_1.plot(ax=axis,label='Forecast',alpha=0.5)
axis.fill_between(TES_Final_1.index, TES_Final_1['lower_CI'],TES_Final_1['upper_CI'],color='k',alpha=.15)
axis.set_xlabel('Year-Month')
axis.set_ylabel('Sparkling')
plt.legend(loc='best')
plt.show();

	Sparkling
count	187.000000
mean	2402.417112
std	1295.111540
min	1070.000000
25%	1605.000000
50%	1874.000000
75%	2549.000000
max	7242.000000

Time_Stamp	April	August	December	February	January	July	June	March	May	November	October	September
Time_Stamp
1980	1712.0	2453.0	5179.0	1591.0	1686.0	1966.0	1377.0	2304.0	1471.0	4087.0	2596.0	1984.0
1981	1976.0	2472.0	4551.0	1523.0	1530.0	1781.0	1480.0	1633.0	1170.0	3857.0	2273.0	1981.0
1982	1790.0	1897.0	4524.0	1329.0	1510.0	1954.0	1449.0	1518.0	1537.0	3593.0	2514.0	1706.0
1983	1375.0	2298.0	4923.0	1638.0	1609.0	1600.0	1245.0	2030.0	1320.0	3440.0	2511.0	2191.0
1984	1789.0	3159.0	5274.0	1435.0	1609.0	1597.0	1404.0	2061.0	1567.0	4273.0	2504.0	1759.0
1985	1589.0	2512.0	5434.0	1682.0	1771.0	1645.0	1379.0	1846.0	1896.0	4388.0	3727.0	1771.0
1986	1605.0	3318.0	5891.0	1523.0	1606.0	2584.0	1403.0	1577.0	1765.0	3987.0	2349.0	1562.0
1987	1935.0	1930.0	7242.0	1442.0	1389.0	1847.0	1250.0	1548.0	1518.0	4405.0	3114.0	2638.0
1988	2336.0	1645.0	6757.0	1779.0	1853.0	2230.0	1661.0	2108.0	1728.0	4988.0	3740.0	2421.0
1989	1650.0	1968.0	6694.0	1394.0	1757.0	1971.0	1406.0	1982.0	1654.0	4514.0	3845.0	2608.0
1990	1628.0	1605.0	6047.0	1321.0	1720.0	1899.0	1457.0	1859.0	1615.0	4286.0	3116.0	2424.0
1991	1279.0	1857.0	6153.0	2049.0	1902.0	2214.0	1540.0	1874.0	1432.0	3627.0	3252.0	2408.0
1992	1997.0	1773.0	6119.0	1667.0	1577.0	2076.0	1625.0	1993.0	1783.0	4096.0	3088.0	2377.0
1993	2121.0	2795.0	6410.0	1564.0	1494.0	2048.0	1515.0	1898.0	1831.0	4227.0	3339.0	1749.0
1994	1725.0	1495.0	5999.0	1968.0	1197.0	2031.0	1693.0	1720.0	1674.0	3729.0	3385.0	2968.0
1995	1862.0	NaN	NaN	1402.0	1070.0	2031.0	1688.0	1897.0	1670.0	NaN	NaN	NaN

y	mean	mean_se	mean_ci_lower	mean_ci_upper
0	1411.356050	397.247686	632.764892	2189.947209
1	1233.062262	407.001539	435.353904	2030.770621
2	1700.192561	408.887836	898.787128	2501.597994
3	1586.559627	410.774963	781.455495	2391.663760
4	1432.380551	412.653828	623.593910	2241.167192

y	mean	mean_se	mean_ci_lower	mean_ci_upper
0	1311.826675	389.899488	547.637721	2076.015628
1	1300.651859	400.572863	515.543475	2085.760243
2	1562.780641	400.600108	777.618856	2347.942426
3	1567.089134	408.005644	767.412766	2366.765502
4	1344.581341	409.042152	542.873455	2146.289228

Q1. Read the data as an appropriate Time Series data and plot the data.¶

Q2. Perform appropriate Exploratory Data Analysis to understand the data and also perform decomposition.¶

Q3.Split the data into training and test. The test data should start in 1991.¶

Q4). Build various exponential smoothing models on the training data and evaluate the model usingRMSE on the test data. Other models such as regression,naïve forecast models and simple averagemodels. should also be built on the training data and check the performance on the test data using RMSE¶

6. Build an automated version of the ARIMA/SARIMA model in which the parameters are selected using the lowest Akaike Information Criteria (AIC) on the training data and evaluate this model on the test data using RMSE.¶

7. Build ARIMA/SARIMA models based on the cut-off points of ACF and PACF on the training data and evaluate this model on the test data using RMSE.¶

8. Build a table (create a data frame) with all the models built along with their corresponding parameters and the respective RMSE values on the test data.¶

9. Based on the model-building exercise, build the most optimum model(s) on the complete data and predict 12 months into the future with appropriate confidence intervals/bands.¶

	YearMonth	Sparkling
0	1980-01	1686
1	1980-02	1591
2	1980-03	2304
3	1980-04	1712
4	1980-05	1471

	YearMonth	Sparkling	Time_Stamp
0	1980-01	1686	1980-01-31
1	1980-02	1591	1980-02-29
2	1980-03	2304	1980-03-31
3	1980-04	1712	1980-04-30
4	1980-05	1471	1980-05-31

	Sparkling
Time_Stamp
1990-08-31	1605
1990-09-30	2424
1990-10-31	3116
1990-11-30	4286
1990-12-31	6047

	Sparkling
Time_Stamp
1991-01-31	1902
1991-02-28	2049
1991-03-31	1874
1991-04-30	1279
1991-05-31	1432

	Sparkling
Time_Stamp
1995-03-31	1897
1995-04-30	1862
1995-05-31	1670
1995-06-30	1688
1995-07-31	2031

	Train_RMSE	Test_RMSE
RegressionOnTime	1279.322346	1389.135175
Naive Model	3867.700802	3864.279352

	Alpha_Values	Train_RMSE	Test_RMSE
0	0.3	1359.511747	1935.507132
1	0.4	1352.588879	2311.919615
2	0.5	1344.004369	2666.351413
3	0.6	1338.805381	2979.204388
4	0.7	1338.844308	3249.944092
5	0.8	1344.462091	3483.801006
6	0.9	1355.723518	3686.794285

	Alpha_Values	Beta_Values	Train_RMSE	Test_RMSE
0	0.3	0.3	1592.292788	18259.110704
1	0.3	0.4	1682.573828	26069.841401
2	0.3	0.5	1771.710791	34401.512440
3	0.3	0.6	1848.576510	42162.748095
4	0.3	0.7	1899.949006	47832.397419
...	...	...	...	...
59	1.0	0.6	1753.402326	49327.087977
60	1.0	0.7	1825.187155	52655.765663
61	1.0	0.8	1902.013709	55442.273880
62	1.0	0.9	1985.368445	57823.177011
63	1.0	1.0	2077.672157	59877.076519

	Alpha_Values	Beta_Values	Train_RMSE	Test_RMSE
0	0.3	0.3	1592.292788	18259.110704
8	0.4	0.3	1569.338606	23878.496940
1	0.3	0.4	1682.573828	26069.841401
16	0.5	0.3	1530.575845	27095.532414
24	0.6	0.3	1506.449870	29070.722592

	Alpha_Values	Beta_Values	Gamma_Values	Train_RMSE	Test_RMSE
0	0.3	0.3	0.3	404.513320	3.927862e+02
1	0.3	0.3	0.4	402.088628	9.513202e+02
2	0.3	0.3	0.5	408.282432	1.470487e+03
3	0.3	0.3	0.6	428.631668	2.181724e+03
4	0.3	0.3	0.7	468.958530	3.513351e+03
...	...	...	...	...	...
507	1.0	1.0	0.6	153394.791826	7.989790e+05
508	1.0	1.0	0.7	94040.964958	1.074413e+06
509	1.0	1.0	0.8	102196.953755	5.010607e+06
510	1.0	1.0	0.9	77924.294413	4.318265e+05
511	1.0	1.0	1.0	239917.432847	1.254280e+05

	Alpha_Values	Beta_Values	Gamma_Values	Train_RMSE	Test_RMSE
0	0.3	0.3	0.3	404.513320	392.786198
8	0.3	0.4	0.3	424.828055	410.854547
65	0.4	0.3	0.4	435.553595	421.409170
296	0.7	0.8	0.3	700.317756	518.188752
130	0.5	0.3	0.5	498.239915	542.175497

	param	AIC
8	(2, 1, 2)	2210.618508
7	(2, 1, 1)	2232.360490
2	(0, 1, 2)	2232.783098
5	(1, 1, 2)	2233.597647
4	(1, 1, 1)	2235.013945
6	(2, 1, 0)	2262.035600
1	(0, 1, 1)	2264.906437
3	(1, 1, 0)	2268.528061
0	(0, 1, 0)	2269.582796

	param	seasonal	AIC
50	(1, 1, 2)	(1, 0, 2, 12)	1555.584247
53	(1, 1, 2)	(2, 0, 2, 12)	1556.076770
26	(0, 1, 2)	(2, 0, 2, 12)	1557.121563
23	(0, 1, 2)	(1, 0, 2, 12)	1557.160507
77	(2, 1, 2)	(1, 0, 2, 12)	1557.340409

	RMSE
ARIMA(0,1,2)	1417.502239
SARIMA(0,1,2)(2,0,2,12)	526.462879
SARIMA(2,1,2)(2,0,2,12)	580.602901

	Train_RMSE	Test_RMSE
Alpha=0.3,Beta=0.4,Gamma=0.3,TripleExponentialSmoothing	404.513320	392.786198
Alpha=0.064,Beta=0.053,Gamma=0.0,TripleExponentialSmoothing	356.782453	463.501976
SARIMA(0,1,2)(2,0,2,12)	NaN	526.462879
SARIMA(2,1,2)(2,0,2,12)	NaN	580.602901
2pointTrailingMovingAverage	NaN	813.400684
4pointTrailingMovingAverage	NaN	1156.589694
Simple Average	1298.483628	1275.081804
6pointTrailingMovingAverage	NaN	1283.927428
9pointTrailingMovingAverage	NaN	1346.278315
ARIMA(2,1,2)	NaN	1374.663870
RegressionOnTime	1279.322346	1389.135175
ARIMA(0,1,2)	NaN	1417.502239
Alpha=0.3,SimpleExponentialSmoothing	1338.805381	1935.507132
Alpha=0.995,SimpleExponentialSmoothing	1372.054747	3855.940897
Naive Model	3867.700802	3864.279352
Alpha=0.3,Beta=0.3,DoubleExponentialSmoothing	1500.689062	18259.110704